COASTAL DYNAMICS MODELLING 2011

LAGRANGIAN DATA ASSIMILATION

La Londe Les Maures, September 22, 2011

Vincent Taillandier (taillandier@obs-vlfr.fr)

MOTIVATIONS

1. Interest on data collected by Lagrangian instruments

Lagrangian instruments are floating buoys moving in good approximation with the ocean current and providing information about their posi-tion.

Davis, R.E., 1985. Drifter observations of coastal surface currents during CODE : the method and descriptive view. Journal of Geophysical Research, 90, 4741-4755.

Davis, R.E., 1991. Observing the general circulation with floats. Deep-Sea Research, 38, 5531–5571.

Davis, R.E., Sherman, J.T., Dufour, J., 2001. Profiling ALACEs and other advances in autonomous subsurface floats. Journal of Atmospheric and Oceanic Technology,

18, 982–993.

Owens, W.B., 1991. A statistical description of the mean circulation and eddy variability in the northwestern Atlantic using SOFAR floats. Progress in Oceanography,

28, 257–303.

2. Great contribution on our understanding of the ocean circulation

Lagrangian instruments have documented the ocean mean flow and variability both at the surface and in the interior.

Fratantoni, D.M., 2001. North Atlantic surface circulation during the 1990s observed with satellite-tracked drifters. Journal of Geophysical Research, 106, 22067–22093.

Griffa, A., R. Lumpkin, M. Veneziani, 2008. Cyclonic and anticyclonic motion in the upper ocean. Geophysical Research Letters, 35, L01608.

Lavander, K.L., Davis, R.E., Owens, W.B., 2000. Direct velocity measurements in the Labrador and Irminger Seas describe pathways of Labrador Sea Water. Nature,

407, 66–69.

Swenson, M.S., Niiler, P.P., 1996. Statistical analysis of the surface circulation of the California Current. Journal of Geophysical Research, 101, 22631–22645.

Testor, P., Gascard, J.C., 2003. Large-scale spreading of deep waters in the Western Mediterranean Sea by submesoscale coherent eddies. Journal of Physical Oceano-

graphy, 33, 75–87.

SOFAR- RAFOScourtesy of Testor and Gascard

SURDRIFT- CODEcourtesy of Rubio and Garreau

ARGOcourtesy of Poulain and Le Traon

Surge in Lagrangian data

Ocean mean flow and variability

DOLCEVITA databaseCOASTAL SCALEcourtesy of Veneziani and Poulain

BASIN SCALEcourtesy of Gerin and Poulain

OBS DENSITY

MEAN FLOW

VARIANCE

3. Integral part of the ocean observing system

Lagrangian instruments are extensively used for monitoring coastal zones, regional areas, or oceanic basins. Because their lifetime can span several months and even years, the oceanic variability can be sampled on a large panel of motion scales.

Poulain, P.M., 2001. Adriatic Sea surface circulation as derived from the drifter data between 1990 and1999. Journal of Marine Systems, 29, 332.

Reverdin, G., Niiler, P.P., Valdimarsson, H., 2003. North Atlantic Ocean surface currents. Journal of Geophysical Research, 108, 3002.

4. Capability for Near Real Time applications

Lagrangian instruments are of primary importance in many relevant applications (search and rescue, forecasting and containment of oil spills or other pollutants) since they can be easily and operationally deployed at reasonable cost, and tracked via remote sensing means.

Castellari, S., Griffa, A., Ozgokmen, T.M., Poulain, P.M., 2001. Prediction of particle trajectories in the Adriatic Sea using Lagrangian data assimilation. Journal of Mar-

ine Systems, 29, 33–50.

Haza, A., A. Griffa, P. Martin, A. Molcard, T.M. Özgökmen, A.C. Poje, R. Barbanti, J. Book, P.M. Poulain, M. Rixen, and P. Zanasca, 2007. Model based directed drift-

er launches in the Adriatic Sea: results from the DART experiment. Geophysical Research Letters, 34, L10605.

Integral part of the ocean observing system

WOCE-typesurface

trajectories

ARGO floats

POSITION OF THE PROBLEM

5. Challenging methodological topic

The core of the problem is the inversion of trajectory data onto velocity fields, related by the non-linear advection equation. Several method-ologies have been developed from ‘pseudo-Lagrangian’ formulations (proxy of the underlying current from successive positions) to Lag-rangian formulations (structure and variability of the underlying current from trajectory properties).

Aref, H., 1984. Stirring by chaotic advection. Journal of Fluid Mechanics, 143, 1-21.

Griffa, A., L.I. Piterbarg, et T.M. Özgökmen, 2004. Predictability of Lagrangian trajectories: effects of uncertainty in the underlying Eulerian flow. Journal of Marine

Research, 62, 135.

6. Statistical approaches: stochastic processes

The advection equation uses a turbulent velocity isolated from the mean flow. This fluctuation is defined by a random walk process in which parameters are estimated from the whole dataset.

Berloff, P., J.C. McWilliams, 2002. Material transport in oceanic gyres. Part 2: hierarchy of stochastic models. Journal of Physical Oceanography, 32, 797-830.

Maurizi, A., Griffa, A., Poulain, P.M., Tampieri, F., 2004. Lagrangian turbulence in the Adriatic Sea as computed from drifter data: effects of inhomogeneity and non -

stationarity. Journal of Geophysical Research, 109, C04010.

Veneziani, M., Griffa, A., Reynolds, A.M., Mariano, A.J., 2004. Oceanic turbulence and stochastic models from subsurface Lagrangian data for the North-West Atlantic

Ocean. Journal of Physical Oceanography, 34, 1884-1906.

7. Global approaches: dynamical systems

The coherent structures of the underlying flow can be identified by the convergence properties of the position dataset or estimated with a best fit of global basis functions with observed velocities.

Poje, A.C., and G. Haller, 1999. Geometry of cross stream mixing in a double gyre ocean model. Journal of Physical Oceanography, 29, 16491665.

Toner, M., Poje, A.C., Kirwan, A.D., Jones, C.K.R.T., Lipphardt, B.L., Grosch, C.E., 2001. Reconstructing basin-scale Eulerian velocity fields from simulated drifter

data. Journal of Physical Oceanography, 31, 1361–1376.

Wiggins, S., 2005. The dynamical systems approach to Lagrangian transport in oceanic flows. Annual Review of Fluid Mechanics, 37, 295328.

8. Local approaches: best linear unbiased estimation

A background velocity field is modified requiring that the distance between observed and model forecasted particle positions is minimized. Such data assimilation approach has been thoroughly tested with positive results using idealized point vortex models, QG and PE models.

Kamachi, M., O’Brien, J.J., 1995. Continuous assimilation of drifting buoy trajectories into an equatorial Pacific Ocean model. Journal of Marine Systems, 6, 159–178.

Kuznetsov, L., Ide, K., Jones, C.K.R.T., 2003. A method for assimilation of Lagrangian data. Monthly Weather Review, 131, 2247–2260.

Molcard, A., Piterbarg, L.I., Griffa, A., Ozgokmen, T.M., Mariano, A.J., 2003. Assimilation of drifter positions for the reconstruction of the Eulerian circulation field.

Journal of Geophysical Research, 108, 3056.

Nodet, M., 2006. Variational assimilation of Lagrangian data in oceanography. Inverse Problems, 22, 245263.

Ozgokmen, T.M., Molcard, A., Chin, T.M., Piterbarg, L.I., Griffa, A., 2003. Assimilation of drifter positions in primitive equation models of midlatitude ocean circula-

tion. Journal of Geophysical Research, 108, 3238.

Taillandier, V., A. Griffa, et A. Molcard, 2006. A variational approach for the reconstruction of regional scale Eulerian velocity fields from Lagrangian data. Ocean

Modelling, 13, 124.

increasing spatial coverage

increasing sampling time

16%

26%

43%

77%

control flow

reconstruction

mesoscale signal + inertial osc

Flow reconstruction

12. Implementation on data assimilation systems

Interests for operational oceanography focused on Argo trajectories whose coarse trajectory sampling is to be accounted over high resolution meshgrids. The Lagrangian method is then implemented as an observation operator to provide a sub-surface information on velocity and as-sociated mass field at the scales specified by background error correlations.

Taillandier V., A. Griffa, P.M. Poulain, K. Béranger. Assimilation of Argo float positions in the north western Mediterranean Sea and impact on ocean circulation

simulations. Geophysical Research Letters, 33, L11604, 2006.

Taillandier, V., A. Griffa, 2006. Implementation of position assimilation from Argo floats in a realistic Mediterranean Sea model and twin experiment testing. Ocean

Science, 2, 223236.

13. “Offline” implementation: flow reconstruction

Another application is to reconstruct outputs of numerical simulations with observed trajectories, in the aim to improve the phasing of circu-lation patterns and the associated transport properties. The optimal velocity increments are superimposed to the background flow over a prior time window with a prior spatial scale.

Taillandier, V., A. Griffa, P.M. Poulain, R. Signell, J. Chiggiato, S. Carniel. Variational analysis of drifter positions and model outputs for the reconstruction of surface

currents in the Central Adriatic during fall 2002. Journal of Geophysical Research, 113, C004148, 2008.

Rubio, A., V. Taillandier, P. Garreau. Reconstruction of the Mediterranean Northern Current variability and associated cross shelf transports in the Gulf of Lions from

satellite tracked drifters and model outputs. Journal of Marine Systems, 2008.

MFSTEP analysissatellite altimetry

ARGO profiles

MFSTEP analysissatellite altimetry

ARGO profilsARGO trajectories

Impact study for MFS (Ligurian Sea)

Impact study on material fluxes (Adriatic Sea)

Residence time upstream the cape

simulated flow estimated flow observed flow

DEVELOPMENT OF THE “BLUE” SOLUTION

9. The minimisation problem

Given an observed drift over a sequence duration characterized by the Lagrangian timescale, the aim is to find the optimal velocity field which identifies this drift under the constraint of the advection equation. Given a cost function measuring the observation misfit, the optimal velocity is searched in the direction of its gradient.

10. Incremental formulation

The incremental formulation is used to approximate this non-linear minimisation problem with a series of quadratic minimisation problems. A background velocity field is then iteratively corrected to identify the observed drift. For each iteration, time-independent velocity incre-ments are estimated under the constraint of the advection equation linearized around the background velocity field.

11. Parameterisation of discrete solutions

The discrete advection equation is parameterised by the number of recurrences and by the spatial resolution of the velocity meshgrid. In case of coarse resolution relative to the sequence drift, velocity increments are obtained over the mesh contained the drift, one recurrence is then sufficient. Otherwise, several recurrences are required to cover the meshes crossed by the trajectory, which then enables a full Lagrangian approach to estimate the observed structure.

EXERCISE

ObjectiveA mesoscale eddy is sampled with a resolution of 6h by three trajectories. Given this position dataset, the aim is to reconstruct the underly-ing velocity field on a meshgrid of resolution 1/16 degree.

DiagnosticThe quality of the reconstruction is sensitive to different parameters that are explored in this exercise. The analysis should be performed dur-ing a experimental window of some days. The solutions can be assessed quantitatively by the Lagrangian predictability of the reconstructed flow, i.e. the misfit between observed trajectories (in red) and simulated trajectories (in blue).

Guide-lines- Pre-processing of a 4 month position dataset (Drifter.dat): (i) characterise the Lagrangian timescale of the observed structure : represent the autocorrelation function of Lagrangian velocities (ii) select an experimental period during which the reconstruction will be performed

- Run the reconstruction method (./oceanS), represent the estimated velocity (output/velest.dat) and trajectories (output/trjest.dat) over the meshgrid (output/meshgrid.dat) (i) explore the sensitivity of the optimisation procedure: sequence duration, number of updates (ii) explore the sensitivity of the prior information: spatial scale, time scale of the reconstructed flow